To design a new base class, it should both be a concept large enough to deserve a full-class design effort, different enough so that it can't easily be represented by an existing class, and generic enough so that it shouldn't be siloed under a specific existing class.
And have a strong enough identity. It's a pretty weird bar, not so much high, as warped.
The Psion could have been a Monk or Sorcerer/Wizard subclass, but it would have carried too much martial or magic baggage respectively.
Or Warlock sub-class, to go with the 'Far Realms' connection. Really, the Sorcerer and Monk have a really hard time justifying themselves under the criteria you set out, above. The Wizard & Fighter, respectively could have handled them pretty easily. (Conversely, a Monk as a sub-class of a psionic Mystic full class wouldn't be too illogical - except for the Monk, like the Barbarian, being more cultural stereotype than archetype.)
The Artificer could have been a Wizard subclass, but it would make it very strong on spells, and thus leaving much less space for actually creating stuff, which is what defines this character concept.
That's an issue homebrewers solved ages ago, about the same time we were coming up with utterly broken 'mana' systems, there were much less prevalent/more obscure systems that let a magic-user make a potion or alchemical item or whatever, and memorize & cast a spell to make it available for the day.
Under 5e spell slots, that'd've been downright intuitive. The Artificer could have learned item formulae as spells and empowered them with slots. Simple.
But the main questions the designers should ask themselves are: should every Artificer actually be a Wizard? Is there enough narrative uniqueness, mechanical difference, and gamer's interest to promote this character concept to a fully-fledged class, or will it always be a niche option that only a few players will ever play? Can it ever hold up besides classes that have been around for decades?
The last question is critical - the Artificer /did/ hold up in at least one edition, and it was iconic to Eberron. So, even though there's little justifying it, letting it slip in, if only for Eberronies, makes sense.
Yes, definitely.
It handles a swath of heroic archetypes that the game has consistently failed for decades. The fighter has long been held up as a leader archetype, the 9th level Lord of the classic game, the 'natural leader who anchors the party' in 3e (cf OotS, for the gentlest mockery of how that went). And the game mechanics have always utterly failed to capture that. And, with good reason: the fighter was too busy being a badass or generic feat-based building-block (or both with enough system mastery).
The 5e fighter is back to the badass archetype and has no design space left to it for the support-style martial leader of warrior (or adventures). Not only that, but many other less overtly heroic (and, well, side-kick) archetypes could also, positively coincidentally at first, use the 4e Warlord's 'leader role' mechanics to finally /work/ in D&D. There's many support-cast roles (common language 'role') you see in genre that aren't badass fighters or cunning/lethal rogues or magic-users of any strip who none the less are important to the story and to their teamates. The 'Lazy' (as charop called them) or as [MENTION=82504]Garthanos[/MENTION] coined the term 'Princess' builds of the Warlord were a way to finally play such concepts without just being a drag on the party.
And that's just the, narrow, limited-by-formal-Roles 4e vision of the Warlord. 5e classes don't have to stick to combat-Role functions, they can be a lot more varied (just look at how much the Cleric and Druid were constrained in 4e compared to CoDzilla in 3.5 - 4e had to cut the Druid into separate Controller and Leader pieces! - vs being off the leash again in 5e). A 5e full Warlord class could potentially do more than just inspire and organize allies in a support mode, it could manipulate enemies, as well, leaning into abilities that would have been denied it in 4e as being 'controller' or 'defender' functions.
On top of that, it's iconic to 4e - and, yes, 5e must maintain the same commitment to fans of 4e as those of other past editions. That alone should have gotten it into the PH. That it was left for later, optional supplements was already a huge 'compromise' to the h4ter side of the fanbase.
The Warden seemed a trifle forced to me, personally, like the Avenger, it seemed like it's schtick was already in use, just under the rubric of a different Role. Both, in 5e, are echoed in the Paladin, but it's just an echo. A Warden might be squeezed in under Druid, though it'd step on the Moon Druids toes (claws).
But, really, the archetype of a warrior who turns into things to defend nature is a little novel. The classic Berserker who turns into a Bear is the closest thing that leaps to mind, and though the Barbarian hasn't gone there this time, it could.
Shaman? ... it would have to be done very well and with some bold design choices, so that it would be clearly different from any other spellcaster.
The D&D take on the divine - the whole 'patron deity' thing - is pretty far removed from the animism of shamans. Though, I suppose, even if there were a shaman full class, it'd be casting many of the same spells as the Cleric, anyway. Considering how crowded the field of spellcasters already is, and how marginally the Sorcerer was differentiated, there's not really a lot of design space left to fill with a Shaman. Even so, it could be worth it. Animism is a major feature of many sources of inspiration for the fantasy genre, and has been mostly under-developed in D&D, being very different from D&D's outer-planar, patron-deity take on the divine. Animistic spirits share the world with the with the Shaman, not meddle with it from beyond. It could be another case where the 5e full-class version would have the opportunity to open up and cover more conceptual design space than it's earlier incarnations.
